PT - JOURNAL ARTICLE
AU -
TI - Coupling coefficients and tensor operators for chains of groups
DP - 1975 Jan 30
TA - Philosophical Transactions of the Royal Society of London. Series A,
Mathematical and Physical Sciences
PG - 545--585
VI - 277
IP - 1272
4099 - http://rsta.royalsocietypublishing.org/content/277/1272/545.short
4100 - http://rsta.royalsocietypublishing.org/content/277/1272/545.full
SO - Philos Trans R Soc Lond A1975 Jan 30; 277
AB - The action of an arbitrary (but finite or compact) group on an arbitrary Hilbert space is studied. The application of group theory to physical calculations is often based on the Wigner-Eckart theorem, and one of the aims is to lead up to a general proof of this theorem. The group’s action gives irreducible ket-vector representation spaces, products of which lead to a definition of coupling (Wigner, or Clebsch-Gordan) coefficients and jm and j symbols. The properties of these objects are studied in detail, beginning with properties that are independent of the basis chosen for the representation spaces. We then explore some of the consequences of choosing bases by using the action of a subgroup. This leads to the Racah factorization lemma and the definition of jm factors, also a general statement of Racah’s reciprocity. In the third part, we add to these ideas, some properties of the space of all linear operators taking the Hilbert space to itself. This leads to a proof of the Wigner—Eckart theorem which is both succinct and in the language of quantum mechanics.